%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% self eivironment                                                            %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\newtheorem{defi}{Definition}
\newtheorem{remark}{Remark}
\newtheorem{thm}{Theorem}
\newtheorem{corollary}[thm]{Corollary}
\newtheorem{question}{Question}
\newtheorem{proposition}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{example}{Example}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% symbols                                                                     %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\usepackage{dsfont}
\newcommand{\I}{\mathds 1}
\newcommand{\real}{\mathbb{R}} 
\newcommand{\complex}{\mathbb{C}}
\newcommand{\natnumber }{\mathbb{N}}
\newcommand{\bbR}{\mathbb{R}}
\newcommand{\bbC}{\mathbb{C}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\iu}{\mathrm{i}\mkern1mu}

\newcommand{\ha}{\mathcal{H}_A} 
\newcommand{\hb}{\mathcal{H}_B} 
\newcommand{\hh}{\mathcal{H}}
\newcommand{\setS}{\mathcal{S}}%
\makeatletter
\newcommand{\notprop}{\propto\kern-1\@ptsize pt \diagup}
\makeatother

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% operators                                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\newcommand{\rank}[1]{\mathrm{rank} (#1)} 
\newcommand{\range}[1]{\mathcal{R} (#1)}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert} 
\newcommand{\kernel}[1]{\mathrm{Ker} (#1)}
\newcommand{\trace}[1]{\mathrm{Tr}(#1)}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\adj}[1]{#1^{\mathsf{H}}}
\newcommand{\trans}[1]{#1^{\intercal}}
\newcommand{\Real}[1]{\mathrm{Re}(#1)}
\newcommand{\Ima}[1]{\mathrm{Im}(#1)}


% \DeclarePairedDelimiter\bra{\langle}{\rvert}
% \DeclarePairedDelimiterX\ket{\lvert}{\rangle}
% \DeclarePairedDelimiterX\braket[2]{\langle}{\rangle}{#1 \delimsize\vert#2}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% abbreviated expression                                                      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\newcommand{\ei}[1][i]{\ket{#1}} % or e_i
\newcommand{\ej}{\ket{j}}
\newcommand{\xx}[1]{\adj{#1}#1}
\newcommand{\sijkl}[2]{S^1_{#1,#2}}
\newcommand{\sijklt}[2]{S^2_{#1,#2}}
\newcommand{\commutes}{\; \substack{\mathrm{commutes}\\
    \xrightarrow{\hspace*{6em}}\\ \mbox{}}\; }
\newcommand\lijkl[2]{\lambda^{1}_{#1 #2}}
\newcommand\lijklt[2]{\lambda^{2}_{#1 #2}}
\newcommand\Lijkl[1]{\Lambda^{1}_{#1 }}
\newcommand\Lijklt[1]{\Lambda^{2}_{#1}}

%% -------------------------------------------------- qed --------------------------------------------------
\renewcommand{\qedsymbol}{$\blacksquare$}

%% -------------------------------------------------- end --------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% abbreviations                                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\makeatletter
\DeclareRobustCommand\onedot{\futurelet\@let@token\@onedot}
\def\@onedot{\ifx\@let@token.\else.\null\fi\xspace}

\def\eg{\emph{e.g}\onedot} 
\def\Eg{\emph{E.g}\onedot}
\def\ie{\emph{i.e}\onedot} 
\def\Ie{\emph{I.e}\onedot}
\def\cf{\emph{c.f}\onedot} 
\def\Cf{\emph{C.f}\onedot}
\def\etc{\emph{etc}\onedot} 
\def\vs{\emph{vs}\onedot}
\def\wrt{w.r.t\onedot} 
\def\dof{d.o.f\onedot}
\def\etal{\emph{et al}\onedot}
\makeatother
\let\strokeL\L
\DeclareRobustCommand{\L}{\ifmmode\mathbf{L}\else\strokeL\fi}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%          renewcommand                                                       % 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\renewcommand{\intercal}{\mathsf{T}}
\renewcommand\tilde[1]{\stackrel{\sim}{\smash{{#1}}\rule{0pt}{1.1ex}}}


%





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